In the art of making paper with modern high-speed machines, sheet properties must be continually monitored and controlled to assure sheet quality and to minimize the amount of finished product that is rejected when there is an upset in the manufacturing process. The sheet variables that are most often measured include basis weight, moisture content, and caliper, i.e., thickness, of the sheets at various stages in the manufacturing process. These process variables are typically controlled by, for example, adjusting the feedstock supply rate at the beginning of the process, regulating the amount of steam applied to the paper near the middle of the process, or varying the nip pressure between calendaring rollers at the end of the process.
A paper machine employs large arrays of actuators spread across a continuously moving web to control the cross-directional (CD) profiles of paper properties as measured by one (or several) scanning sensor(s) downstream from the actuators. The CD actuators generally consist of an array of independently controlled actuators mounted in a beam that spans the width of the moving sheet in the CD. Traditionally, CD actuators have been designed with the intent of controlling a single measured profile. Slice lip and dilution profiling actuators are located in the headbox and are designed to control the weight profile of the paper. Steam box and rewet shower actuators are located further down the machine and are designed to control the moisture content of the sheet. Finally, induction heating actuators are located at the dry end of the paper machine and locally heat the rolls in the calendar stack and through thermal expansion or contraction of the roll to increase or decrease the pressure imposed on and, hence the caliper of the paper sheet.
In practice these actuators often have a significant effect on more than one measured profile. For example, slice lip actuators are force actuators which are designed to modify the height of the gap exiting the headbox and, thus, control the distribution of stock extruded on the wire screen in an effort to control the weight profile. Data from newsprint machines have shown that operation of the slice lip also significantly impacts the moisture profile (as opening slice means more water on wire and slower dewatering for heavier sheets). Additional multivariable effects arise from the use of “redundant” actuator arrays.
CD control refers to the control system designed to reduce the variability in the paper sheet properties as a function of the cross-direction. Typically, designers are using pairing rules to choose one CD actuator array for controlling one paper sheet property and the interaction of multiple array CD processes is usually neglected in traditional CD control.
Most well-designed single array CD systems are unfortunately ill-conditioned. Even at steady-state, some of their singular values are vanishingly small. The large dimensionality and the ill-conditioning make these processes challenging to control. It has been recently found that for multiple array CD processes the ill-conditioning of the process could be due to the interaction between multiple array measurements and actuators. That means it can be much more difficult to control multiple array CD systems than single array CD systems.
Application of model predictive control (MPC) in CD processes has been considered for some time. Although most published papers consider only one actuator array and one controlled property and consequently do not address the problem of coordinating multiple CD actuator arrays controlling multiple sheet properties, multiple array CD control systems are becoming more prevalent. Industrial model predictive control implementation can employ a multiple-array model of the CD process that is obtained from a complementary industrial model identification tool such as the method described in U.S. Pat. No. 6,086,237 to Gorinevsky and Heaven which is assigned to Honeywell International, Inc. The advantages of multiple-array control are evident in the improved performances that have been reported. The main disadvantage is the enormous computational load required for online optimization as the constrained quadratic programming (QP) problem may be required to generate as many as 600 actuator setpoints subject to up to 1800 constraints from up to 6000 measurements as often as every 15 seconds.
A procedure for implementing a paper machine CD MPC control system is shown as a sequence of six steps in FIG. 1. The tuning step where the prediction horizon and control horizon, and optimization weights are selected is often ad hoc and typically evaluated via simulations of the closed-loop system. A consistent automated tuning method for large-scale CD MPC is described in U.S. patent application Ser. No. 11/260,809 entitled “Automated Tuning of Large-Scale Multivariable Model Predictive Controllers for Spatially-Distributed Process,” by Fan & Stewart and filed on Oct. 27, 2005.
Even after the multivariable CD predictive controller is properly tuned, the task of correctly predicting the controller's performance, which corresponds to step 4 in FIG. 1, in an efficient way with the optimal tuning parameters in the face of active constraints, remains. This has not been accomplished satisfactorily especially for dynamic performance prediction because of the large scale dimensional problem. The state of the art of performance prediction for large-scale MPC with active constraints is to run the MPC in a closed-loop simulation that may take at least 10 to 15 minutes to complete.